Conjugate of Dirichlet Character

This just follows because $z^{-1}=\bar{z}$ if $|z|=1$. That is, negating the argument of a complex number and flipping it about the real axis are the same geometric action.

Proof

Since $gcd(a,q)=1$, we can consider $\chi$ as a group character when restricted to ${\mathbb{Z}}_q^{\ast }$ by identifying $a$ with the corresponding reduced residue class. Then because characters of finite group map to roots of unity, we know that $\chi (a)$ is a root of unity, and hence $\chi (a{)}^n=1⟹|\chi (a){|}^n=1⟹|\chi (a)|=1$.

Thus writing $\chi (a)=e^{i\theta }$ we can see that